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On-line secret sharing

机译：在线秘密分享

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In a perfect secret sharing scheme the dealer distributes shares to participants so that qualified subsets can recover the secret, while unqualified subsets have no information on the secret. In an on-line secret sharing scheme the dealer assigns shares in the order the participants show up, knowing only those qualified subsets whose all members she has seen. We often assume that the overall access structure (the set of minimal qualified subsets) is known and only the order of the participants is unknown. On-line secret sharing is a useful primitive when the set of participants grows in time, and redistributing the secret when a new participant shows up is too expensive. In this paper we start the investigation of unconditionally secure on-line secret sharing schemes. The complexity of a secret sharing scheme is the size of the largest share a single participant can receive over the size of the secret. The infimum of this amount in the on-line or off-line setting is the on-line or off-line complexity of the access structure, respectively. For paths on at most five vertices and cycles on at most six vertices the on-line and offline complexities are equal, while for other paths and cycles these values differ. We show that the gap between these values can be arbitrarily large even for graph based access structures. We present a general on-line secret sharing scheme that we call first-fit. Its complexity is the maximal degree of the access structure. We show, however, that this on-line scheme is never optimal: the on-line complexity is always strictly less than the maximal degree. On the other hand, we give examples where the first-fit scheme is almost optimal, namely, the on-line complexity can be arbitrarily close to the maximal degree. The performance ratio is the ratio of the on-line and off-line complexities of the same access structure. We show that for graphs the performance ratio is smaller than the number of vertices, and for an infinite family of graphs the performance ratio is at least constant times the square root of the number of vertices.
机译：在完美的秘密共享方案中，交易者将股份分配给参与者，以便合格的子集可以恢复秘密，而不合格的子集则不包含有关秘密的信息。在在线机密共享方案中，发牌人按照参与者出现的顺序分配股份，只知道她所见过的所有成员的合格子集。我们经常假设整体访问结构（最小合格子集）是已知的，而仅参与者的顺序是未知的。当一组参与者随时间增长时，在线秘密共享是一个有用的原语，而当新的参与者出现时重新分配秘密太昂贵了。在本文中，我们开始研究无条件安全的在线秘密共享方案。秘密共享方案的复杂性是单个参与者可以接收的最大共享大小超过秘密大小。在线或离线设置中此金额的最小值分别是访问结构的在线或离线复杂性。对于最多五个顶点的路径和最多六个顶点的循环，在线和离线复杂度相等，而对于其他路径和循环，这些值不同。我们表明，即使对于基于图的访问结构，这些值之间的差距也可以任意大。我们提出了一种通用的在线机密共享方案，称为“最适合”。它的复杂性是访问结构的最大程度。但是，我们证明了这种在线方案从来都不是最优的：在线复杂度始终严格小于最大程度。另一方面，我们给出的例子中，首次拟合方案几乎是最优的，即在线复杂度可以任意接近最大程度。性能比是相同访问结构的在线和离线复杂度之比。我们表明，对于图来说，性能比小于顶点数，而对于无穷多个图，性能比至少是常数乘以顶点数的平方根。

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